681 research outputs found
On the computational complexity of the abelian permutation group structure, membership and intersection problems
AbstractAlgorithms on computations on abelian permutation groups are presented here. An algorithm for computing the complete structure, algorithms for membership-inclusion testing and an algorithm for computing the intersection of abelian permutation groups are given. Their worst-case time complexity is a polynomial of degree 4 in terms of n, the number of points moved by the group. The upper bounds on the running time of the algorithms shown here improve the bounds on the above problems cited in the literature
Parallel algorithms for solvable permutation groups
AbstractA number of basic problems involving solvable and nilpotent permutation groups are shown to have fast parallel solutions. Testing solvability is in NC as well as, for solvable groups, finding order, testing membership, finding centralizers, finding centers, finding the derived series and finding a composition series. Additionally, for nilpotent groups, one can, in NC, find a central composition series, and find pointwise stabilizers of sets. The latter is applied to an instance of graph isomorphism. A useful tool is the observation that the problem of finding the smallest subspace containing a given set of vectors and closed under a given set of linear transformations (all over a small field) belongs to NC
An Efficient Quantum Algorithm for some Instances of the Group Isomorphism Problem
In this paper we consider the problem of testing whether two finite groups
are isomorphic. Whereas the case where both groups are abelian is well
understood and can be solved efficiently, very little is known about the
complexity of isomorphism testing for nonabelian groups. Le Gall has
constructed an efficient classical algorithm for a class of groups
corresponding to one of the most natural ways of constructing nonabelian groups
from abelian groups: the groups that are extensions of an abelian group by
a cyclic group with the order of coprime with . More precisely,
the running time of that algorithm is almost linear in the order of the input
groups. In this paper we present a quantum algorithm solving the same problem
in time polynomial in the logarithm of the order of the input groups. This
algorithm works in the black-box setting and is the first quantum algorithm
solving instances of the nonabelian group isomorphism problem exponentially
faster than the best known classical algorithms.Comment: 20 pages; this is the full version of a paper that will appear in the
Proceedings of the 27th International Symposium on Theoretical Aspects of
Computer Science (STACS 2010
The Parallel Dynamic Complexity of the Abelian Cayley Group Membership Problem
Let be a finite group given as input by its multiplication table. For a
subset of and an element the Cayley Group Membership Problem
(denoted CGM) is to check if belongs to the subgroup generated by .
While this problem is easily seen to be in polynomial time, pinpointing its
parallel complexity has been of research interest over the years. In this paper
we further explore the parallel complexity of the abelian CGM problem, with
focus on the dynamic setting: the generating set changes with insertions
and deletions and the goal is to maintain a data structure that supports
efficient membership queries to the subgroup . We obtain the
following results:
1. We first consider the more general problem of Monoid Membership. When
is a commutative monoid we give a deterministic dynamic algorithm
constant time parallel algorithm for membership testing that supports
insertions and deletions in each step.
2. Building on the previous result we show that there is a dynamic randomized
constant-time parallel algorithm for abelian CGM that supports
polylogarithmically many insertions/deletions to in each step.
3. If the number of insertions/deletions is at most
then we obtain a deterministic dynamic constant-time parallel algorithm for the
problem.
4. We obtain analogous results for the dynamic abelian Group Isomorphism
The counting complexity of group-definable languages
AbstractA group family is a countable family B={Bn}n>0 of finite black-box groups, i.e., the elements of each group Bn are uniquely encoded as strings of uniform length (polynomial in n) and for each Bn the group operations are computable in time polynomial in n. In this paper we study the complexity of NP sets A which has the following property: the set of solutions for every x∈A is a subgroup (or is the right coset of a subgroup) of a group Bi(|x|) from a given group family B, where i is a polynomial. Such an NP set A is said to be defined over the group family B.Decision problems like Graph Automorphism, Graph Isomorphism, Group Intersection, Coset Intersection, and Group Factorization for permutation groups give natural examples of such NP sets defined over the group family of all permutation groups. We show that any such NP set defined over permutation groups is low for PP and C=P.As one of our main results we prove that NP sets defined over abelian black-box groups are low for PP. The proof of this result is based on the decomposition theorem for finite abelian groups. As an interesting consequence of this result we obtain new lowness results: Membership Testing, Group Intersection, Group Factorization, and some other problems for abelian black-box groups are low for PP and C=P.As regards the corresponding counting problem for NP sets over any group family of arbitrary black-box groups, we prove that exact counting of number of solutions is in FPAM. Consequently, none of these counting problems can be #P-complete unless PH collapses
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